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\leftline{Ty,}
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Thanks for being a good sport and taking all of my suggestions for your oral 
presentation to heart.
You did a good job there and I think people got a better idea of what algebraic
statistics is about by the example you did of the $3\times 3$ independence model
and the MLE for the mixture model.   
Unfortunately, the paper just is not at that same level because there are a 
number of things you either got more or less wrong or did not explain very well. 
You also didn't include the second example from the talk, which would have
made a really nice addition!

In the classes you 
have taken with me, I have noticed a consistent pattern in how you approach
learning mathematics along the lines of what I think happened with this paper.
This is something you will need to change
if you want to do anything related to this subject in the future.  What I mean is that
you too often ``settle for'' superficial understanding of topics or subjects without
really making yourself work out all of the details or forcing yourself to 
think things through carefully.  See the comment 7 below for the biggest single example
of this from the paper.  But that's far from the only example. 
I have seen the same thing on problem sets, 
in oral problem presentations, on the midterm exam, and so forth.  When I have
seen students with similar issues in the past, it has often been because
they didn't really care and just couldn't be bothered to put in the time 
to really learn the material.  I DON'T think that is the case with you, though,
so I have to say that I'm somewhat mystified why this keeps happening in 
the work you submit.  (Could it be that math always came easily for you
in the past and you never had to do this before?) 
The thing I will say is that I think you need to learn
to be more self-critical and hold yourself to a higher standard.  I wouldn't
say this if I wasn't convinced you {\it had the ability and motivation} to 
do that.

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\noindent
{\it Specific Comments}
\bigskip
\item{1.}  I guess you are thinking of thinks like the Captcha tests built
in to lots of website logins.  That doesn't seem like such a big inconvenience, though.
And if it adds a layer of protection against bot attacks on websites that
have information I want to keep private (like credit card numbers, etc.) I'm
willing to put up with them.
\item{2.}  This is not very precise.  I think you are trying to suggest 
the properties of Markov chains that you introduce two paragraphs ahead.
But it would be better to wait and be more careful there.
\item{3.}
It would have been good to reorganize what you say and include
some more details about the relation 
between what is often called the JC {\it rate matrix}:
$$Q = \pmatrix{-3\alpha & \alpha & \alpha & \alpha\cr
            \alpha & -3\alpha & \alpha & \alpha\cr
            \alpha & \alpha & -3\alpha & \alpha\cr
            \alpha & \alpha & \alpha & -3\alpha\cr}$$
and the substitution
matrix from page 4.  They are equivalent in a sense, but I think
it's probably better to start from the rate matrix $Q$.  The idea is
that $\theta(t)$ has to be a solution of the differential 
equation $\theta'(t) = Q\cdot \theta(t)$ with $\theta(0) = I$ (identity matrix).
This is a more precise way of saying that Jukes-Cantor is based on 
the assumption that the time evolution is a Markov process.  Any continuous-time
memoryless dynamical system looks like this.  Then, by general properties of 
solutions of linear constant-coefficient ODE, $\theta(t)$ is given by the 
matrix exponential:
$$\theta(t) = e^{Qt},$$ 
and $\theta(t)$ is the matrix in the equation on your page 4.  
The point though, is that the entries
in $\theta(t)$ are the transition probabilities per unit time $\mu_i$
and $\pi_i$ from before (you need the $i$'s if you're looking at more
than one site in the genome, but otherwise, they could be omitted as 
below).  Hence
$$\mu = {1\over 4}\left(1 + 3 e^{-4\alpha}\right)$$
and 
$$\pi = {1\over 4}\left(1 - e^{-4\alpha}\right)$$
from setting $t = 1$ in the formula for $\theta(t)$.
\item{4.}  The subscripts in the $\mu_i$ and $\pi_i$ allow the transition probabilities
to be different in different locations along the DNA sequence.  
In the basic Jukes-Cantor model, they are the same for all transitions 
between nucleotides at any particular site as shown
in the diagrams from page 6.  You only need the subscript $i$'s if you 
are talking about different sites in the genome.  
Different nucleotides (A,C,G,T) {\it are not the same thing as different sites}.  
To talk about the {\it sites} in DNA sequences, there has to a way to align those 
sequences and identify one pair of nucleotides in one sequence with one pair of
nucleotides in another sequence.
\item{5.}  I wouldn't say the Poisson distributions are ``uniquely reliable.''
They are a standard way to model the number of occurrences of some event within
limited regions in continuous time and/or space.  
\item{6.}  There is a typo here -- the condition in the first term 
should be $rt = C$, not $rt = A$.
\item{7.}  I'm afraid you really misunderstood the example from David Cox's 
survey that you are working from here.  
The thing you didn't notice or didn't understand is that this
is supposed to be a {\it ``binary''} Jukes-Cantor model.  That means
that $\pi_0,\pi_1$ are just binary data: $0$ or $1$.  {\it In other words, this whole 
discussion $\underline{does\ not\ apply}$ directly to non-binary data like the $A,T,G,C$ nucleotides in DNA.}  That's why the transition matrices on the three branches are just
$2\times 2$ matrices and not $4\times 4$ matrices.  Also, this set-up
is different from the Jukes-Cantor DNA model because notice that the 
three transition matrices {\it could be different}.  The discussion
of the variety corresponding to this statistical model, following Cox, is good.  
The problem is that it just doesn't apply to the situation you are studying in the 
rest of the paper.  The thing that is parallel in the DNA situation is the derivation of 
the ``condensed'' Jukes-Cantor model from pages 150 and 151 of 
the Pachter-Sturmfels book.  I don't think you thought about this carefully
enough to really understand what was going on and recognize that what you said
could not apply to the DNA evolution model.
\item{8.}  The expected number of mutations over a time period $t$ is $3\alpha t$
as you say.  But since you have set up all the equations, it would also
have been good to say more about the actual derivation of the 
JC distance (or ``correction'').  The idea is that starting from any
mother (ancestor) species, the probability of a change after time $t$
is the sum of the three off-diagonal terms in the corresponding 
row of the $\theta(t)$ matrix:  ${3\over 4}(1 - e^{-4\alpha t})$.
So if you see that $k/n$ of the positions in a sequence of the 
genome have changed, then the maximum likelihood estimate for
the branch length is obtained just by solving the equation 
$${k\over n} = {3\over 4}(1 - e^{-4\alpha t})$$
for $-4\alpha t$, then multiplying that by ${-3\over 4}$ to get 
the $3\alpha t$:
$${4k\over 3n} = 1 - e^{-4\alpha t},$$
so 
$$-4\alpha t = \ln\left(1 - {4k\over 3n}\right)$$
and then multiply both sides by ${-3\over 4}$.  
\bigskip
\noindent
Final Project Presentation:  92 (A-)
\bigskip
\noindent
Final Project Paper:  82 (B-) 
\bye
